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Modelowanie Nanostruktur
Lecture 4 1
Modelowanie Nanostruktur
Semester Zimowy 2011/2012
Wykład
Jacek A. Majewski
Chair of Condensed Matter Physics
Institute of Theoretical Physics
Faculty of Physics, Universityof Warsaw
E-mail: [email protected]
Modelowanie Nanostruktur, 2011/2012
Jacek A. Majewski
Wykład 4 – 25 X 2011
Continuous Methods for Modeling Electronic Structure of Nanostructures Effective Mass Approximation
Modeling Nanostructures
Nanotechnology –
Low Dimensional Structures
Quantum
Wells
Quantum
Wires
Quantum
Dots
A B
Simple
heterostructure
Modelowanie Nanostruktur
Lecture 4 2
What about realistic nanostructures ?
2D (quantum wells): 10-100 atoms in the unit cell
1D (quantum wires): 1 K-10 K atoms in the unit cell
0D (quantum dots): 100K-1000 K atoms in the unit cell
Organics
Nanotubes, DNA: 100-1000 atoms (or more)
Inorganics
3D (bulks) : 1-10 atoms in the unit cell
TEM image of a InAs/GaAs dot Si(111)7×7 Surface
GaN
InGaN
GaN
HRTEM image:
segregation of Indium
in GaN/InGaN
Quantum Well
Examples of Nanostructures
Synthesis of colloidal nanocrystals
Injection of organometallic
precursors
Mixture of surfactants Heating mantle
Nanostructures: colloidal crystals
-Crystal from sub-µm
spheres of PMMA
(perpex) suspended in
organic solvent;
- self-assembly when
spheres density high
enough;
illuminated with white light
Bragg„s law different
crystals different orientation
different
Modelowanie Nanostruktur
Lecture 4 3
Hot topic (to come) –
The curious world of nanowires
Self-organized growth of nanowires:
catalytic VLS growth
Mechanizm Vapor-Liquid-Solid
VLS (Wagner 1964)
SEM of ZnTe nanowires grown by MBE
on GaAs with Au nanocatalyst
Nanowire site control and branched NW structures:
nanotrees and nanoforests
(A) Nanowires can be accurately positioned using lithographic methods such as EBL and NIL. (B) Subsequent seeding by aerosol
deposition produces nanowire branches on an array as in panel (A). Shown here is a top view of such a ‘nanoforest’ where the
branches grow in the <111>B crystal directions out from the stems. (C) Dark field STEM image and EDX line scan of an
individual nanotree. An optically active heterosegment of GaAsP in GaP has been incorporated into the branches.
http://www.nano.ftf.lth.se/
Modelowanie Nanostruktur
Lecture 4 4
Nanowire nanolasers
Room temperature lasing action from chemically synthesized
ZnO nanowires on sapphire substrate
Huang, M., Mao, S.S., et al., Science 292,
1897 (2001)
Schematic illustration A SEM image
One end of the nanowire is the epitaxial interface between the
sapphire
and ZnO, whereas the other end is the crystalline ZnO (0001)
plane
exposed in air.
Detection of single viruses with NW-FET
Patolsky & Lieber, Materials Today, April 2005, p. 20 Patolsky et al., Proc. Natl. Acad. Sci.
USA 101 (2004) 14017
Simultaneous conductance and optical data recorded
for a Si nanowire device after the introduction of influenza A solution
Controlled Growth and Structures of
Molecular-Scale Silicon Nanowires
(a) TEM images
of 3.8-nm SiNWs
grown along the
<110> direction
(c) cross-sectional
image
(b) & (d) models
based on
Wulff
construction
Yue Wu et al., NANO LETTERS 4, 433
(2004)
High Performance Silicon Nanowire
Field Effect Transistors
Yi Cui, et al. NANO LETTERS 3, 149
(2003)
Comparison of SiNW FET
transport parameters
with those for
state-of-the-art planar
MOSFETs show that
“SiNWs have the
potential to exceed
substantially conventional
devices, and thus could be
ideal building blocks for future
nanoelectronics.”
Modelowanie Nanostruktur
Lecture 4 5
Heterostructured Nanowires
coaxial
heterostructured
nanowire
longitudinal
heterostructured
nanowire
Heterostructured Nanowires
Transmission electron microscopy images of
a GaN/AlGaN
core–sheath nanowire
two Si/SiGe
superlattice nanowires
Simulation methods
Atomistic methods for modeling of
nanostructures
Ab initio methods (up to few hundred atoms)
Modelowanie Nanostruktur
Lecture 4 6
Density Functional Theory (DFT)
One particle density determines the ground state energy
of the system for arbitrary external potential
extE [ ρ] d r ρ( r )υ ( r ) F [ ρ] 3
E [ ρ ] E0 0
ground state density
ground state energy
ext S x cE[ ρ] drυ ( r )ρ( r ) T [ ρ] U [ ρ] E [ ρ] E [ ρ]
unknown!!!
Total energy
functional
External
energy
Kinetic
energy
Classic Coulomb
energy
Exchange
energy
Correlatio
n
energy
Atomistic methods for modeling of
nanostructures
Ab initio methods (up to few hundred atoms)
Semiempirical methods (up to 1M atoms)
Tight-Binding Methods
Tight-Binding Hamiltonian
† †iα iα iα iα , jβ iα jβ
αi αi ,βj
H ε c c t c c
creation & anihilation operators
On-site energies are not atomic eigenenergies
They include on average the effects
of neighbors
Problem: Transferability
E.g., Si in diamond lattice (4 nearest neighbors)
& in fcc lattice (12 nearest neighbors)
Dependence of the hopping energies on the distance
between atoms
0 2 6 4 8 10 12 14 1
10
100
1 000
10 000
100 000
1e+06
Nu
mb
er
of
ato
ms
R (nm)
Tight-Binding
Pseudo-
potential
Ab initio
Atomistic vs. Continuous Methods
Microscopic approaches can be applied
to calculate properties of realistic nanostructures
Number of atoms in a spherical Si nanocrystal as a function of its radius R.
Current limits of the main techniques for calculating electronic structure.
Nanostructures commonly studied experimentally
lie in the size range 2-15 nm.
Continuous
methods
Modelowanie Nanostruktur
Lecture 4 7
Atomistic methods for modeling of
nanostructures
Ab initio methods (up to few hundred atoms)
Semiempirical methods (up to 1M atoms)
(Empirical Pseudopotential)
Tight-Binding Methods
Continuum Methods
(e.g., effective mass approximation)
Continuum theory-
Envelope Function Theory
Electron in an external field
2ˆ( ) ( ) ( ) ( )
2
pV r U r r r
m
Periodic potential of crystal Non-periodic external potential
Strongly varying on atomic scale Slowly varying on atomic scale
0
-5
5
1
3
1
1
1
3
„2
„25
15
1
„2
„2
5
1
5
L „2
L1
L „3
3L
L14
1
En
erg
y [
eV
]
Wave vector k
11
Ge
Band structure
of Germanium
( )n
k0U( r )
Band Structure
Electron in an external field 2ˆ
( ) ( ) ( ) ( )2
pV r U r r r
m
Periodic potential of crystal Non-periodic external potential
Strongly varying on atomic scale Slowly varying on atomic scale
Which external fields ?
Shallow impurities, e.g., donors
Magnetic field B,
Heterostructures, Quantum Wells, Quantum wires, Q. Dots
2
( )| |
eU r
r
B curlA A
GaAs GaAlAs
cbb
GaAs GaAlAs GaAlAs
Does equation that involves the effective mass and a slowly varying
function exist ? 2ˆ( ) ( ) ( )
2 *
pU r F r F r
m
( ) ?F r
Modelowanie Nanostruktur
Lecture 4 8
Envelope Function Theory –
Effective Mass Equation J. M. Luttinger & W. Kohn, Phys. Rev. B 97, 869 (1955).
[ ( ) ( ) ] ( ) 0ni U r F r
0( ) ( ) ( )n nr F r u r
( ) 0U r ( ) exp( )nF r ik r
(EME)
EME does not couple different bands
Envelope
Function
Periodic
Bloch Function
“True”
wavefunction
Special case of constant (or zero) external potential
( )r Bloch function
( )U z ( ) exp[ ( )] ( )n x y nF r i k x k y F z
Electronic states in
Quantum Wells
Envelope Function Theory- Electrons in Quantum Wells
Effect of Quantum Confinement on Electrons
Let us consider an electron in the conduction band near point
GaAs GaAlAs GaAlAs
cbb
Growth direction (z – direction )
Potential ? ( )U r
( )U r is constant in the xy plane
22
0 *( )
2c ck k
m
GaAs
GaAlAs
0
0
( ) ( )c
c c
zU r U z
E z
2 2 2 2
2 2 2( , , ) ( ) ( , , ) ( , , )
2 *F x y z U z F x y z EF x y z
m x y z
Effective Mass Equation for the Envelope Function F
( , , ) ( ) ( ) ( )x y zF x y z F x F y F zSeparation Ansatz
2 222
2 2 2( )
2 *
y zxy z x z x y x y z x y z
F FFF F F F F F U z F F F EF F F
m x y z
2 222
2 2 2( )
2 *
y zxy z x z x y x y z x y z
F FFF F F F F F U z F F F EF F F
m x y z
x y zE E E E
22
22 *
xy z x x y z
FF F E F F F
m x
22
22 *
yx z y x y z
FF F E F F F
m y
22
2( )
2 *
zx y x y z z x y z
FF F U z F F F E F F F
m z
22 2
2
2~ ,
2 * 2 *xik xx
x x x x x
FE F F e E k
m mx
Effective Mass Equation of an Electron in a Quantum Well
22 22
2~ ,
2 * 2 *
yik yyy y y y y
FE F F e E k
m my
22
2( )
2 *
znzn zn zn
FU z F E F
m z
Modelowanie Nanostruktur
Lecture 4 9
Conduction band states of a Quantum Well
|||| || ||
1 1( ) exp[ ( )] exp( )x yk
F r i k x k y ik rA A
|| |||| || ||,
1( ) ( ) ( ) exp( ) ( )zn znn k k
F r F r F z ik r F zA
22
|| ||( )2 *
n znE k k Em
Energies of bound states
in Quantum Well
22
2( )
2 *
znzn zn zn
FU z F E F
m z
En
erg
y
n=1
n=2
n=3
0c
0c cE znE ||( )nE k
||k
22
1 ||2 *
E E km
22
2 ||2 *
E E km
1E
2E
Wave functions Fzn(z)
znE
( )znF z
Band structure Engineering
The major goal of the fabrication of heterostructures is the
controllable modification of the energy bands of carriers
Lattice constant
En
erg
y G
ap
[e
V]
Band lineups in semiconductors
-12
-10
-8
-6
-4
0
InS
b
GaS
b
InA
s
AlS
b
GaA
s
AlA
s
InP
GaP
AlP
InN
GaN
AlN
Zn
O
Zn
S
Cd
S
Zn
Se
EN
ER
GY
(e
V)
Band edges compile by: Van de Walle (UCSB) Neugebauer (Padeborn), Nature’04
e.g., GaAs/GaAlAs InAs/GaSb
Band lineup in GaAs / GaAlAs
Quantum Well with Al mole fraction
equal 20%
Envelope Function Theory- Electrons in Quantum Structures
Type-I Type-II
vbt
cbb A
B
vE
cE
AgapE
BgapE
vbt
cbb
A B
vE
cE
AgapE
BgapE
vbt
cbb A
B vE
cE
AgapE
BgapE
Various possible band-edge lineups in heterostructures
GaN/SiC
(staggered) (misaligned)
- Valence Band Offset (VBO) vE - Conduction band offset cE
GaAlAs vbt
cbb
1.75 eV 1.52 eV
0.14 eV
0.09 eV
GaAs
VBO’s can be only obtained either from experiment or ab-initio calculations
Modelowanie Nanostruktur
Lecture 4 10
Density of states
We define to refer to a single direction of electron spin
(denoted )
Density of States
G(E)dE - the number of electron states with energies E and E + dE
Generally, this quantity is proportional to the volume of the crystal
Convention: we define G to refer to the volume of a unit cell
3
3( ) ( ( ))
(2 )n
n
G E d k E k
G
Of course, there is no contribution from
a particular band at energy E unless
there are states of energy E in that band
Band index
2 2
*( ) (0)
2n n
n
kk
m
3/ 2(3 )
2 2
* 2( )
D mG E E
Density of States of a Two-Dimensional Electron Gas
A special function known as the density of states G(E) that gives
the number of quantum states dN(E) in a small interval dE around
energy E: dN(E)= G(E) dE
-the set of quantum numbers (discrete and continuous)
corresponding to a certain quantum state
( ) ( )G E E E
Energy associated with
the quantum state
||{ , , }s n k For 2DEG:
Spin quantum number Continuous two-dimensional vector
A quantum number characterizing the
transverse quantization of the electron states
22 2
, ,
( ) 2 [ ( )]2 *
x y
n x y
n k k
G E E E k km
Density of States of a Two-Dimensional Electron Gas 2
2 2
, ,
( ) 2 [ ( )]2 *
x y
n x y
n k k
G E E E k km
,x yL L - are the sizes of the system in x and y directions
x yS L L - the surface of the system 2
,
( ) ( )(2 )
x y
x yx y
k k
L Ldk dk
22 2
2
22
|| || ||2 0
2|| || ||2 0
( ) 2 [ ( )]2 *(2 )
2 ( )2 *2
2 *( )
x yx y n x y
n
x yn
n
x yn
n
L LG E dk dk E E k k
m
L Lk dk E E k
m
L L mk dk E E k
2|| ||k
|| ||2 20
* *( ) ( ) ( )n n
n n
Sm SmG E d E E E E
( )x - Heaviside step function for and for ( ) 1 0 ( ) 0 0x x x x
Modelowanie Nanostruktur
Lecture 4 11
Density of States of a Two-Dimensional Electron Gas
2
*( ) ( )n
n
SmG E E E
Often the density of states per unit area,
, is used to eliminate the size
of the sample
( ) /G E S
Each term in the sum corresponds to the contribution from one subband.
The contributions of all subbands are equal and independent of energy.
The DOS of 2DEG exhibits a staircase-shaped energy dependence,
with each step being associated with one of the energy states.
De
ns
ity o
f s
tate
s
1E 2E 3E E
(2 )( )
DG E
3/ 2(3 )
2 2
* 2( )
D mG E E
(3 )( )
DG E
Density of states for 2DEG in an
infinitely deep potential well
22
*( )
2k k
m
For large n, the staircase
function lies very close to
the bulk curve (3 )
( )D
G E
2
*m
2
2 *m
2
3 *m
Density of States of a Two-Dimensional Electron Gas
Bulk
300 A QW 100 A QW
0 100 200
De
ns
ity o
f s
tate
s
Energy [meV]
Dependence on the width of Quantum Well
QW of large width Bulk
Electronic states in
Quantum Wires
Electron States in Quantum Wires
xk
Free movement in the x-direction,
Confinement in the y, z directions
( , , ) ( , )xik xnF x y z e F y z
2 2 2
2 2( , ) ( , ) ( , ) ( , )
2 *n n n nF y z U y z F y z E F y z
m y z
22
( )2 *
n x n xE k E km
2 2(1 )
,
( ) 2 ( )2 *
x
D xn
n k
kG E E E
m
(1 )
2
2 * 1( ) ( )
D xn
n n
L mG E E E
E E
( , )U y zConfinement potential
Density of states for one-dimensional electrons
De
ns
ity o
f s
tate
s
1E 2E 3E E
Modelowanie Nanostruktur
Lecture 4 12
Electronic states in
Quantum Dots
Electron States in Quantum Dots
A
B
A
Self-organized quantum dots
Electrons confined
in all directions
( , , )U x y z
2 2 2 2
2 2 2( , , ) ( , , ) ( , , ) ( , , )
2 *n n n nF x y z U x y z F x y z E F x y z
m x y z
(0 )( ) ( )
DG E E E
Density of states for zero dimensional (0D) electrons (artificial atoms)
De
nsity o
f sta
tes
1E 2E 3E E4E
Confinement Quantization of energy levels
L
L
L
How small should be semiconductor system to see
separate levels in room temperature?
x x
y y
z z
k L πn
k L πn
k L πn
2
2
2
x y z
πE )
m* L( n n n
22 2 22
2
2
ΔE
BΔE k K . eV 300 0 026
m* m* m 0
nmL .m*
2 21173 55
GaAs m* . 0 03 L nm 76
Si m* . 0 2 L nm 29
Electronic states in
Nanostructures
Summary
Modelowanie Nanostruktur
Lecture 4 13
Energy spectrum
1D: free motion in x direction, dimensional quantisation in z,y
[H,px] = 0
m,n(r) = uk(r)eikxxFm,n(y,z)/(Lx)1/2
2D: free motion in x,y directions,dimensional quantisation in z
[H,px] = 0; [H,py] = 0
n(r) = uk(r)ei(kxx+kyy)Fn(z)/(Lx Ly)1/2
3D: free motion in x,y,z directions
[H,p] = 0
k(r) = uk(r)eikr /(Lx Ly Lz)1/2
Density of States of Electrons in
Semiconductor Quantum Structures
A A B Quantum Wells
A B A
B Bulk
Quantum Wires
Quantum Dots
3D
2D
1D
0D
(2 )( )
DG E
1E 2E 3E E
2
*m
2
2 *m
2
3 *m
(3 )( )
DG E
E
1E 2E 3E E
(1 )( )
DG E
1E 2E 3E E4E
(0 )( )
DG E
3/ 2(3 )
2 2
* 2( )
D mG E E
(2 )
2
*( ) ( )
Dn
n
mG E E E
(0 )( ) ( )
Dn
n
G E E E
(1 )
2 2
( )2 *( )
D n
n n
E EmG E
E E
22
0 *( )
2c ck k
m
B A
A
Position Dependent
Effective Mass
Effective Mass Theory with Position Dependent
Electron Effective Mass
* *A Bm m
*Am *
Bm
0z
2 2
22 * ( )
d
m z dz
21
2 * ( )
d d
dz m z dz
21
( ) ( ) ( ) ( )2 * ( )
d dF z U z F z EF z
dz m z dz
ˆ ˆ[ * ( )] [ * ( )] [ * ( )]z zT m z p m z p m z 2 1
( )F z1 ( )
* ( )
dF z
m z dz
0z
*Am
*( )Bm z
“Graded structures”
IS NOT HERMITIAN !!
Symetrization of the kinetic energy operator
General form of the kinetic energy operator
with
IS HERMITIAN !
and ARE CONTINOUOS !
Modelowanie Nanostruktur
Lecture 4 14
Doping in Semiconductor
Low Dimensional Structures
vE
cE
Energy band diagram of a selectively doped AlGAAs/GaAs
Heterostructure before (left) and after (right) charge transfer
AlGaAs
GaAs
VACUUM LEVEL
vE
EF
AgE
BgEA
B
AgE
1
dl
ALNegatively
charged
region
Positively charged
region
A Band - The electron affinities of material A & B
The Fermi level in the GaAlAs material is supposed to be pinned
on the donor level.
The narrow bandgap material GaAs is slightly p doped.
BgE
(0)( ) ( ) ( )U z U z e z
Effects of Doping on Electron States in Heterostructures
+ +
Ec
+ + EF + + +
+ + + Ec (z) EF
1E
Unstable Thermal equilibrium Charge transfer
2 2
( ) ( )
4( ) ( ) ( ) | ( ) |A D
acc don
er r R r R f r
Resulting electrostatic potential
should be taken into account in the Effective Mass Equation
2 2 2 2
2 2 2( , , ) ( ) ( ) ( )
2 *U x y z e r r E r
m x y z
2 24( ) ( ) ( ) | ( ) |A D
er N r N r f r
Electrostatic potential can be obtained from the averaged acceptor and
donor concentrations
Fermi distribution function
The self-consistent problem, so-called “Schrödinger-Poisson” problem
Thank you!